Chemical Engineering Science 60 (2005) 2557–2570

www.elsevier.com/locate/ces

Application of a three-layer modeling approach for solids transport inhorizontal and inclined channels

A. Ramadana,∗, P. Skalleb, A. Saasenc

aDepartment of Petroleum Engineering, University of Tulsa, NCDB, 2450 E. Marshall St., Tulsa, OK 74110, USAbDepartment of Petroleum Engineering and Applied Geophysics, NTNU, Trondheim, Norway

cStatoil , N-4035 Stavanger, Norway

Received 10 May 2003; received in revised form 27 August 2004; accepted 5 December 2004

Abstract

The three-layer model concept developed previously for solid–liquid flow has been adapted to model solids transport in inclined channels.The present model predicts the pressure loss and transport rate of solids in Newtonian and power-law fluid suspensions by assumingstratified flow conditions. Sets of stationary sand bed transport rate tests were performed to verify the predictions of the model. A 70-mmflow loop was constructed to measure the average transport rates and critical flow rates, which are required to initiate the motion of solidsbed particles. The tests were carried out by eroding stationary sand beds with water and an aqueous solution of poly anionic cellulose(PAC) in a transparent pipe. Four sand beds with different particle size ranges were used. The average transport rates of the beds werepredicted using the model. The model predictions show a satisfactory agreement with experimentally measured results when the grainReynolds number is between 15 and 400 and the flow rate is sufficiently higher than the critical flow rate. Therefore, with some degreeof limitation, the three-layer model can be applicable for predicting the transport rates of stationary solids beds in inclined channels forboth Newtonian and power-law fluids.� 2005 Elsevier Ltd. All rights reserved.

Keywords:Modeling; Non-Newtonian fluid; Moving bed; Particle; Slurries; Suspension; Solid transport

1. Introduction

Suspension flows are known to display substantiallynon-uniform concentration distributions, which influencehydraulic characteristics of the flow. Many engineeringproblems involve suspension flows in ducts of complexgeometry. During drilling of inclined wells, drilling fluidcirculates through the annular space between the drill pipeand the borehole. The recirculation is mainly to transportrock cutting from the well to the surface. Often accuratepredictions of cutting transport rate and frictional pressureloss are necessary in the design of the drilling process. Esti-mating the cutting transport rate requires accurate hydraulicpredictions and knowledge of cutting concentration profile

∗ Corresponding author. Tel.: +1 9186315174; fax: +1 9186315009.E-mail address:ramadan-ahmed@utulsa.edu(A. Ramadan).

0009-2509/$ - see front matter� 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2004.12.011

in the annulus. Nonetheless, obtaining a simple solution forthe concentration profile is difficult. As a result, differentmodeling approaches are proposed to predict the concentra-tion profile, transport rate and frictional pressure loss.

Since the introduction of directional wells, considerableefforts have been made to solve the cutting transport prob-lem in highly inclined and horizontal wells. For good cuttingtransport, vertical flow is preferable because cutting fall inthe opposite direction to the velocity of the drilling fluid(mud). For an inclined well, the direction of cutting settlingis still vertical, but the fluid velocity has a reduced verticalcomponent. This decreases the fluid’s capability to suspenddrilled cuttings. At a high angle of inclination (when theinclination is measured from vertical), a cutting particle thatslips through the mud has a short distance to travel beforestriking the borehole wall. Once it has reached the wall,the particle has little chance of entrainment because local

2558 A. Ramadan et al. / Chemical Engineering Science 60 (2005) 2557–2570

Flow direction

Pipe movement

Fig. 1. Sticking of drill pipe during tripping operation.

(a) (b) (c)

Fig. 2. Typical borehole geometric configurations in inclined wells.

fluid velocities near the wall are very low and insufficientto re-entrain the particle into the flow. The most commondrilling problems related to cutting transport are either insuf-ficient hole cleaning or excessive annular velocity that leadsto borehole erosion and fracturing of the formation. Usu-ally insufficient hole cleaning results in a buildup of cuttingconcentration in the annulus, consequently sticking of thepipe, lost circulation and hindering the running of a casing.Fig. 1 shows how cutting beds cause problems when thedrillstring is moved axially during a tripping operation (Rasi,1994). Unlike vertical wells, the flows in inclined wells arenot annular due to the formation of cutting bed on the lowside of the borehole. As a result, borehole configurations ininclined wells vary with cutting bed height, eccentricity ofthe drill pipe and diameters of the borehole and drill pipe.The most common types of borehole geometric configura-tions in inclined wells are presented inFig. 2.

During drilling of both vertical and inclined wells, non-Newtonian fluids such as clay (Bentonite), oil-based mudsand polymer-viscosified fluids are often used as a drillingfluid. An aqueous solution of poly anionic cellulose (PAC)is one of the commonly used polymer-viscosified drillingfluids. Experimental and modeling studies suggested that,in addition to the flow rate, the rheology determines cuttingtransport capacity of drilling fluids. Non-Newtonian fluidproperties such as shear-thinning and thixotropic behaviors(yield stress) are normally considered in selecting a drilling

fluid that has higher cutting transport ability with optimumfrictional pressure loss. As a result, in practical field appli-cations, non-Newtonian fluid properties such as yield stress,consistency index and Power-law exponent are routinelycontrolled on the surface along with the flow rate (pumprate). Commercially available hydraulic and cutting trans-port models are used to predict the effects of drilling fluidproperties and other drilling parameters. However, modelpredictions need improvement to minimize hydraulics andcutting transport-related problems. Often such problems canlead to expensive and costly operations. Therefore, a cut-ting transport model that is applicable for non-Newtonianfluid is essential to obtain reasonable hydraulic and cuttingtransport predictions.

In order to model stratified solid–liquid flow in pipes,Wilson (1976) developed a one-dimensional two-layermodel that assumes flow of two layers (phases). Thesolid–liquid mixture in each layer is considered as a singlephase. Each layer has a separate velocity, with momentumtransfer between the layers due to shear forces at their in-terface. This approach has been extended by a number ofworkers includingGavignet and Sobey (1989)who pre-sented the two-layer cutting transport modeling concept.The two-layer modeling approach assumes that the lowerpart of the inclined wellbore forms a bed that slides upthe annulus together with a layer of suspension. In thisapproach, two mechanisms can be expected to affect thecutting bed. One of the mechanisms is saltation that occurswhen the drag force is exerted by the liquid phase on aparticle resting at the interface. The drag force initiates themovement of the bed particle. The other one is sliding thattakes place when the shear stress exerted by the liquid overthe rough surface and the pressure gradient result in slidingof the entire bed by overcoming the solid–solid friction atthe wall. However, the two-layer modeling approach ignoresthe most important phenomenon that occurs near the bed.

Bagnold (1954)experimentally found the existence of dis-persive shear stress. Dispersive shear stress originates fromthe interchange of momentum, which is caused by inter-particle collision and hydrodynamic interference (FredsZeand Deigaard, 1992; Davis, 1996). In addition to the fluidshear stress, the dispersive shear stress is responsible for theentrainment of stationary bed particles. The two-layer mod-eling approach neglects the presence of the dispersed layer.Therefore, this assumption reduces the accuracy and applica-bility of the model, because the physical phenomena occur-ring near the bed is different from the other part of the flow.Experimental observation indicates the existence of a highlyconcentrated thin layer of suspension near the bed. The con-cept of dispersive layer appears to have a phenomenologicalbackground. As a result, the suspension above a solids bedis considered to have two different fluid layers with distinctconcentration profiles.

The concept of dispersive layer has been employed byDoron and Barnea (1993)to extend the two-layer modelingapproach to a three-layer scheme. Their model considered

A. Ramadan et al. / Chemical Engineering Science 60 (2005) 2557–2570 2559

c

y

0.5

τs

τdb

τiFdb Fd τd

Fb τb

θb θd

Ss

SiSdb

Sd

Sb

Solids Bed

Suspended Layer

Dispersed Layer

(a) (b)

Fig. 3. (a) Schematic representation of shear stresses acting in thethree-layer mechanics model; and (b) assumed concentration profiles inthree-layer modeling scheme.

the existence of a dispersive layer, which is sandwiched be-tween the suspended layer and the bed as shown inFig. 3a.The dispersed layer was considered to have a higher concen-tration gradient compared to the suspended layer (Fig. 3b).No-slip condition between the solid particles and the fluidwas assumed. This assumption is applicable when the flowis in horizontal or near-horizontal configurations. The modelpredictions showed satisfactory agreement with experimen-tal data, which were obtained in horizontal pipes.Nguyenand Rahman (1998)have adopted the three-layer model forcutting transport predictions. They have studied the effectof various geometric and flow parameters on the hydraulicsand transport rate of cuttings. Recently,Cho et al. (2002)have proposed a new approach to overcome the limitationsof the existing three-layer model and adopted the model forinclined channel flows. Nonetheless, the model neglects thesettling actions of the particles, which may significantly af-fect the model predictions.

Although some efforts have been expended on develop-ing the three-layer cutting transport model, its applicationis still limited to Newtonian fluid flows in horizontal andnearly horizontal channels. In the present study, the three-layer model is extended for use in non-Newtonian inclinedchannel flows. The no-slip assumption between the particlesand the fluid, which was the shortcoming of the previousmodeling studies, has been eliminated. The no-slip assump-tion is often used to simplify the model; however, it restrictsthe model to horizontal and near-horizontal configurations.

2. Model development

The three-layer model presented in the study is developedto overcome the limitations in the existing models, whichare used to predict cutting transport in inclined and horizon-tal wells. Material balance equations of the two phases andmomentum equations of the three layers are combined todevelop the model. Additional equations are introduced toestimate the average concentration of the suspended layer,and thickness and velocity of the dispersed layer. The thick-ness of the dispersed layer is modeled using the pseudo-

hydrostatic pressure gradient concept and assuming linearlyvarying particle concentration in the dispersed layer. As pro-posed byWilson (1987), the velocity profile of the dispersedlayer is estimated based on the turbulent mixing length the-ory. The convection–diffusion equation is used to determinethe average concentration of the suspended layer. In orderto compare the model predictions with the measured data(measured average transport rate), transport rate predictionswere performed at different bed heights and average trans-port rates were determined.

2.1. Model assumptions

Like other modeling procedures, the three-layer model hasits own idealizations and assumptions that are necessary torepresent the physical model in a series of simple mathemat-ical relationships between the flow parameters and proper-ties of the fluid and solid particles. Therefore, the followingassumptions are made to develop the present model:

(i) Distinct imaginary interface lines are to exist betweenthe dispersed layer and suspended layer, and betweenthe dispersed layer and the bed.

(ii) Uniform layers are present without significant variationin concentration and thickness along the length of thechannel.

(iii) The relative velocity between the particles and the fluidis negligible in the bed.

(iv) The flow is steady and turbulent.(v) Bed shear stress variation in the lateral direction is neg-

ligible.(vi) Stratified and well-compacted bed to resist the applied

shear.

Nguyen and Rahman (1998)mentioned that there are var-ious transport modes depending on a given set of operatingconditions. They considered five modes of transport in theirmodel. Based on visual observation of a flow loop test,Fordet al. (1990)categorized the transport process into differenttransport modes, which are presented inFig. 4. Accordingly,Figs. 4a and b show sub-critical conditions where the flowvelocity is not strong enough to lift the top-most particles ofthe bed layer into the dispersed layer. Furthermore, at a lowangle of inclination (�) the gravitation force, which is actingdownhill, becomes high enough to overcome the pressuregradient and the shear stress on the bed. Consequently, thebed begins to slide down as shown inFig. 4a. Otherwise,the bed becomes stationary.

At higher flow rates (mean fluid velocities), the intensityof the local velocity at the bed becomes strong enough to en-train the particles. As a result, the dispersed and suspendedlayers will form with three different bed layer sliding pos-sibilities as shown inFigs. 4c–e. Accordingly, the bed layermay slide down or slide up or be stationary during the trans-port. If the bed layer does not slide at higher flow veloci-ties, then it gets thinner and eventually disappears, leaving

2560 A. Ramadan et al. / Chemical Engineering Science 60 (2005) 2557–2570

Fig. 4. Possible modes of solids transport in inclined channel.

behind the dispersed and suspended layers as shown inFig.4f. Finally, further increase in the flow velocity suspends thebed particles and forms a fully suspended flow as shown inFig. 4g.

2.2. Material balance

With the above assumptions, the material balance equa-tions of solid and liquid phases for stratified solid–liquidflows in inclined channels can be written as

(us − vs cos�)csAs + (ud − vd cos�)cdAd + ubcbAb= (uave− vave cos�)caveAt = Tr (1)

and

us(1 − cs)As + ud(1 − cd)Ad + ub(1 − cb)Ab= uave(1 − cave)At , (2)

respectively, whereu is the mean fluid flow velocity in thelayers,v the particle settling velocity in the layers,A thecross-sectional area of the layers,c the average volumetricconcentration of solids particles in the layers, andTr thetransport rate of the solids. The subscripts “s”, “ d” and “b”refer to the suspended layer, dispersed layer and bed layer,respectively.

2.3. Momentum balance

For the stationary bed condition, the average settling ve-locity of the particles in the stream,vave, can be estimatedby applying settling momentum balance for solids particles.Thus, the total settling momentum equals the sum of settlingmomentum of particles in each layer. Since the settling ve-locity in the bed is negligible, the average settling velocity

equation after simplification becomes

vave= vscsAs + vdcdAdcsAs + cdAd + cbAb . (3)

Here, it is worthwhile to give the physical meaning ofterms in Eqs. (1) and (2). The terms in the right-hand side ofEq. (1) represent the solid transport rates in each layer, whilethe terms in the right-hand side of Eq. (2) represent the fluidflow rates in each layer. These material balance equationscannot yield the solution by themselves; therefore, we needto include the force balance equations as well. Under steady-state condition, the sum of all forces acting on a singlelayer must be zero. Thus, the momentum equation for thesuspended layer becomes

−As dp

dx− �sSs − �iSi − �sAsg cos� = 0, (4)

where� is the angle of inclination from vertical as shownin Fig. 4a, dp/dx the pressure gradient,�i is the interfacialshear stress between the suspended and the dispersed layers,Si is the perimeter in contact between these layers as shownin Fig. 3a and�s is the interstitial shear stress acting on theperimeterSs .

The shear stress at the interface can be estimated by adopt-ing the most widely used Darcy–Weibach equation. Accord-ingly, the interfacial shear stress is given by

�i = 18�sfi(Us − Ud)2, (5)

whereUs andUd are mean flow velocities of the suspendedand dispersed layers, respectively. The friction coefficient atthe interface,fi , can be obtained using the friction factorcorrelations. However, it is necessary to multiply the frictioncoefficient by a factor of 2, to account for the dispersivestress at the interface (Doron and Barnea, 1993). The friction

A. Ramadan et al. / Chemical Engineering Science 60 (2005) 2557–2570 2561

factor correlation, which is applicable for power-law fluids,is given bySzilas et al. (1981)as

1√f

= −2 log

(10−�/2

Re f (2−n)/2n + dp

3.71Dh

), (6)

where� is given by

� = 1.511/n(

0.707

n+ 2.12

)− 4.015

n− 1.057, (7)

wheredp denotes the mean diameter of the particles. Thehydraulic diameter of the layer isDh andn is the flow be-havior index. The density of the suspended layer,�s , is esti-mated as�s = cs�p + (1− cs)�L, where�p and�L are thedensity of solids particles and the fluid, respectively. Similarsuspension density formulas can be used for the other lay-ers. Eq. (6) is applicable for both smooth and rough walls.The generalized Reynolds number for power-law fluids iscalculated as

Re = 8�sU2−ns Dnh

Ks

(6n+ 2

n

)−n, (8)

whereKs is the suspended layer consistency index, whichis roughly estimated byKs=K(1+2.5cs) (Einstein, 1906).Likewise, the interstitial fluid shear stress,�s , in Eq. (4) canbe estimated as�s=fs�su2

s /8, wherefs is the friction factorbetween the fluid and the channel, which can be determinednumerically from Eq. (6).

Furthermore, the momentum balance of the dispersedlayer can be written as

− Ad dp

dx+ �iSi − (�dSd + Fd)

− (�dbSdb + Fdb)− �dAdg cos� = 0, (9)

where�d is the fluid shear stress between the suspensionlayer and the channel acting on the perimeter,Sd . The fluidshear stress between the dispersion layer and the bed,�db,acts on the perimeter,Sdb. The dispersive force,Fdb, in-cludes the inter-particle collision and hydrodynamic inter-ference forces that act on the interface,Sdb. For the sake ofsimplicity, dispersive force,Fdb, and other forces in the mo-mentum equations are considered in unit length basis. Basedon the above definition, an equivalent dispersive shear stresscan be estimated as�dis = Fdb/Sdb.

The dry friction force,Fd , is acting on the boundariesof Sd . This force is estimated using a pseudohydrostaticpressure distribution. Thus, the approximate equation for thedry friction force is given by

Fd ≈ g�d sin �(�p − �L)cdSdhd cos

(�b + �d

2

), (10)

where�d is the dry dynamic friction coefficient between theparticles and the wall of the channel andg is the gravitationalacceleration. The thickness of the dispersed layer ishd . Theangular bed thicknesses (Fig. 3a) of the bed and dispersedlayer are�b and�d , respectively.

The fluid shear stress at the interface between the bed andthe dispersed layer is estimated by

�db = 18�dfdb(Ud − Ub)2, (11)

wherefdb is the interfacial fluid friction factor that can be es-timated using Eq. (6). However, in this case, we do not needto multiply the factor by two, as suggested byTelevantoset al. (1979), because the dispersive force,Fdb, takes careof the dispersive shear stress. The interstitial shear stress,�d , can be evaluated in a similar way to the interstitial shearstress,�s . However, the hydraulic diameter of the dispersedlayer can be approximated by the hydraulic diameter of thesuspended layer. Direct calculation of the hydraulic diameterfrom the flow area and the perimeter of the dispersed layerdoes not give a reasonable friction factor, because this layeris at least in contact with one moving layer, sharing almost50% of its perimeter. As a result, the use of directly calcu-lated hydraulic diameter overestimates the pressure loss andthe interfacial shear stress.

Similarly, the momentum equations of the bed can bewritten as

− Ab dp

dx+ (�dbSdb + Fdb)− (�bsb + Fb)

− �bAbg cos� = 0, (12)

where�b is the interstitial shear stress between the bed layerand the channel, andSb is the perimeter in contact with thechannel. The dry friction force,Fb, acts on the boundarySb. The interstitial fluid shear stress between the bed layerand the channel,�b, can be evaluated in a similar way to theinterstitial fluid shear,�s .

Theoretically upward sliding of the bed occurs from thestationary condition whenFb >Fst, whereFst is the staticfriction force acting on the surface of the channel. Similarly,if Fb <−Fst, then the bed will slide downward as shown inFig. 4c. During upward sliding of the bed, the magnitude ofthe dry friction force becomes that of the dynamic frictionforce, Fdyn. Similarly for downward slidingFb = −Fdyn.The static friction force and the dynamic friction force canbe evaluated byFst = Fn�s andFdyn = Fn�d , respectively,whereFn is the normal force and�s the static friction factorbetween the bed and the channel. The normal force canbe estimated by integrating the pseudohydrostatic pressuredistribution over the boundary of the bed in contact with thechannel. Thus, the local normal force acting on a differentialelement on the surface of the channel can be calculated as

dFn = g(�p − �L)cb

(D

2(cos� − 1)+ h

)

×(D

2d� cos�

)sin �, (13)

whereh is the bed height and� the angular displacement(Fig. 5). The total normal force can be obtained by integrat-ing Eq. (13) from−�b to �b. Subsequently, the simplifiedform of the total normal force acting on the surface of the

2562 A. Ramadan et al. / Chemical Engineering Science 60 (2005) 2557–2570

θb

dθ

h-(r-rcos(θ))

r-rcos(θ)

h

θ

Fig. 5. Geometric considerations in the determination of the pseudohy-drostatic pressure distribution in the bed layer.

channel becomes

Fn ≈ gDcbu(�p − �L)(0.5D(0.5�bu + 0.25 sin 2�bu− 0.5 sin �bu)+ h sin �bu) sin �. (14)

2.4. Thickness and velocity of dispersed layer

In order to complete the three-layer modeling, additionalmodel equations are necessary for determining the followingunknowns: (i) average concentration of the suspended layer;(ii) thickness of dispersed layer; and (iii) velocity of thedispersed layer. Often the thickness of the dispersed layeris determined using the pseudohydrostatic pressure gradientconcept and assuming linearly varying particle concentrationdistribution in the dispersed layer (FredsZe and Deigaard,1992). Thus, the average concentration of dispersed layer isapproximately given by

cd ≈ cs + cb2

. (15a)

In drilling operations, average suspended layer concentra-tion, cs , is very small compared to the average concentra-tion of dispersed layer. Therefore, for the iterative numericalprocedure, the value ofcd can be initialized as 0.5cb.

After adopting for the three-layer model approach, theformula for dispersed layer thickness can be estimated as

hd = �dis

g sin � cb(�P − �L) tan �D, (15b)

where tan(�D) is an equivalent dynamic friction factor.Bagnold (1954)experimentally showed that the value oftan(�D) is around 0.75 near the bed. In naturally formedbeds of solids the concentration of solids is usually taken tobe 0.5. Thus, Eq. (15b) can be rewritten in a simple form as

hd = 2.67�dis

g sin �(�P − �L). (16)

Based on the turbulent mixing length theory,Wilson(1987) proposed the following relation to determine thevelocity profile of the dispersed layer relative to the bed(FredsZe and Deigaard, 1992):

Ud,r (y)= U�

�2

√2y

hd. (17)

The von Karman constant,�, is 0.4 and the distance fromthe mean bed level isy. The friction velocity,U�, is given by(�i/�s)

0.5. An expression for the mean relative velocity canbe obtained by averaging the valueUd,r over the thicknesshd . Thus, the final expression for absolute mean velocity ofthe dispersed layer is

Ud = 4√

2

3�

√�i�P

+ Ub. (18)

2.5. Average concentration of suspended layer

At this stage, the average concentration of the suspendedlayer is unknown.Wilson and Pugh (1988)suggested thatthe concentration in this layer is negligible. However,Doron (1994)has used a convection–diffusion model ap-proach to determine the local concentration (csl) profile inthis layer. Consequently, a one-dimensional time depen-dent, convection–diffusion equation for an inclined channelbecomes

�csl

�t= �

�y

[

�csl

�y+ vscsl sin �

], (19)

where is the solid particles diffusion coefficient. The firstterm in the bracket is the diffusive term that represents theupward flux of solid particles by the action of turbulent ed-dies or other hydrodynamic interactions and the second oneis the connective term, which represents the sedimentationflux of the particles due to the gravitational force. In orderto fulfill the steady-state condition, the sum of terms in thebracket must be zero, yielding the steady-state convectiondiffusion equation as

�csl

�y= −vScsl sin �. (20)

For a rectangular duct, assuming constant settling velocityand diffusivity coefficient, the average concentration of thesuspended layer can be obtained from Eq. (20). Thus, the av-erage concentration of the suspended layer is approximatelyestimated by

cs ≈ cd1 − e−(vs/) sin � hs

vs sin �hs, (21)

wherehs is the thickness of the suspended layer. Obviously,Eq. (21) is not an exact equation of the average concentra-tion, but it gives a reliable approximate value. At this point,it is important to note that similar analysis may not be appli-cable for the dispersed layer, because in the dispersed layer,

A. Ramadan et al. / Chemical Engineering Science 60 (2005) 2557–2570 2563

the normal component of the settling velocity,vd , is bal-anced not only by the hydrodynamic diffusion but also bythe particle collision. As a result, the use of Eq. (20) for thedispersed layer can be inappropriate. Based on experimentalinvestigation,FredsZe and Deigaard (1992)have suggestedthe use of linearly varying particle concentration profile forthe dispersed layer.

Theoretical and experimental studies on particle disper-sion in turbulent flows indicate that the diffusivity of solidparticles is roughly equal to or greater than the diffusiv-ity of the fluid. This assumption is supported by theoreticalresults of homogeneous isotropic turbulence (Mohs et al.,2000). Moreover, the Reynolds analogy between mass andmomentum transport suggests that fluid diffusivity is thesame as eddy viscosity (Young and Leeming, 1997). There-fore, the solid particle turbulent diffusivity coefficient in thesuspended layer can be estimated as = �U�(1 − y/hs)y.This means that the diffusivity coefficient has a parabolicprofile with zero value at the bottom of the suspended layer.In order to avoid zero diffusivity value, the coefficient is usu-ally evaluated a small distance from the interface betweenthe suspended and dispersed layers.FredsZe and Deigaard(1992)have recommended this distance to be twice the meanparticle diameter of a solids bed. In the present study, thisdistance is assumed to behd/100. The laboratory experi-ments indicated that the thickness of the dispersed layer andthe diffusivity of the suspended layer increase with the flowvelocity.

Determination of the settling velocity of the particles isnecessary in Eqs. (1), (3) and (21). Settling has a negativeimpact on solid transport and it depends on the properties ofthe fluid and suspended particles. Generally, a solid particlefalling in a fluid under the action of gravity accelerates untilthe buoyancy and drag force just balance the gravitationalforce, and then it continues to fall at constant velocity, whichis given by (Dedegil, 1987)

v =√

4gdp(�p − �f )

3�LCD, (22)

whereCD is the particle drag coefficient that is determinedby (White, 1974):

CD = 24

Rep+ 6

1 + Re0.5p

+ 0.4, (23)

whereRep is the particle Reynolds number. The above equa-tion can be valid for Newtonian and non-Newtonian fluidsif the definition of the particle Reynolds number is the samein both cases. Hence, it is necessary to define the particleReynolds number in a more general form as (Dedegil, 1987)

Rep = �Lv2s

�, (24)

where� is the shear stress, which is determined by the rheo-logical model of the fluid at representative shear ratevs/dp.The effect of concentration on settling velocity needs to be

considered when calculating the free terminal settling ve-locity of a single particle, because its effect is significant athigh concentrations. Therefore, settling velocity obtained byEq. (22) has to be modified by the hindered settling factorto account for hydrodynamic interference and particle col-lision. For solids volume fractions between 0.001 and 0.4,this factor is given asfS = e−5.9c (Govier and Aziz, 1972).

2.6. Overall momentum and material balance

Finally, the solution for a given mode of transport is de-termined by solving these sets of equations simultaneously.However, the numerical calculation procedure can be sim-pler for a sliding bed case, if the overall momentum andmass balance equations are expressed as

UsAs + UdAd + UbAb = UaveAt , (25)

− At dpdx

− (�sss)− (�bsb + Fdyn)

− (�sAs + �bAb)g cos� = 0, (26)

whereUs , Ud , andUb are the mean flow velocity of thelayers. The average velocity of the stream,Uave, is the ratioof the total flow rate of the suspension to the total cross-section area of the channel. A general formula for the meanflow velocities of the layers can be written as

Ui = Qi

Ai= ui − vici cos�, (27)

whereQi andAi are the flow rate and flow cross-sectionalarea of a given layer, respectively. The subscripti can beany of the subscripts (s, d, b or ave) used in Eqs. (1) and (2).At high inclination angles (i.e., cos� ≈ 0), the followingassumptions can be made:Us=us ;Ud=ud ; andUb=ub. Forincompressible flow, the total flow rate across the channelis given by

Q=Qs +Qd +Qb. (28)

When a stationary bed slides the relative velocity betweenthe bed and the suspended layer approaches the critical ve-locity and results in the formation of a very thin dispersedlayer. Hence, the dry friction forces and the fluid shear stressforces acting between the channel wall and the dispersedlayer can be neglected in the derivation of Eq. (26).

2.7. Average transport rate

To compare the model prediction with the measured data,we need to predict the average transport rate of solids.Eq. (1) only determines the transport rate at a given bedheight or sand volume. The average transport rate is esti-mated by taking the weighted average as

Tave= 1

Vs

∫ 0

Vs

Tr dV , (29)

2564 A. Ramadan et al. / Chemical Engineering Science 60 (2005) 2557–2570

whereV denotes the volume of sand during the test thatranges from the initial sand volume,Vs to zero.

3. Computation procedures and modes of transport

A system of 11 equations ((2), (4), (9), (12), (15a), (16),(18), (21) and (27) for each layer) with 11 unknowns (cs , cd ,Us , Ud , Ub, us , ud , ub, hd , �dis, and dp/dx) can be estab-lished, when all the constitutive relations are substituted intothe conservation equations. This system is non-linear andrequires an iterative procedure to get a numerical solution.Determining the mode of cutting transport and obtaining astable numerical procedure are the most important parts ofthe three-layer modeling. In order to determine the mode oftransport, the numerical procedure needs some criteria forthe range of flow parameters. This requires in-depth under-standing of the model equations and their physical interpre-tations. Thus, it is essential to see the influence of modes oftransport on the key flow parameters such as pressure-dropand flow rate.

Fig. 4 presents possible modes of solids transport in in-clined channels. At low flow rates, the flow velocity is notsufficient enough to initiate the movement of the bed parti-cles; and a two-layer flow pattern shown inFig. 4a will beformed. The two-layer flow pattern has a stationary bed ofsolids and clear liquid layer. However, as the flow velocityincreases, the intensities of hydrodynamic forces grow instrength and eventually reach a stage at which these forcesare strong enough to lift the topmost particles of the bed.At this stage, the three-layer flow pattern with a dispersedlayer establishes as shown inFig. 4c–e. Therefore, in themodel, the thickness of a dispersed layer (hd ) can be usedto determine the presence of the dispersed layer or the tran-sition from a two-layer to three-layer flow pattern. As theflow velocity further increases, the dispersed layer will getthicker. In this condition, the intensity of the hydrodynamicand dispersive shear stresses at the interface between thebed and the dispersed layer may become strong enough toslide the stationary bed. The sliding of the bed occurs whenthe dry friction force (Fb) is greater than the static frictionforce (Fst). If the bed does not slide at a higher flow veloc-ity, then the bed layer will get thinner and eventually disap-pear. The flow pattern changes to a two-layer flow pattern(suspended and dispersed layers) as shown inFig. 4f. Fur-ther increase in the flow velocity will suspend all the bedparticles and form a fully suspended flow pattern (Fig. 4g).However, in its present form, the model cannot predict thetransition from a three-layer flow pattern to a two-layer or afully suspended flow pattern.

In addition to the dispersed layer thickness (hd ) and dryfriction force (Fb), the pressure loss can give some infor-mation about the transition from stationary bed to slidingbed flow condition. In order to see the effect of sliding onthe hydraulics, the frictional pressure loss is estimated us-ing the classical two-layer model approach for sliding bed

Flowrate

Pre

ssur

e dr

op

Stationary bedSliding bed

D

A

qmin

C

B

E

qmax

F

qs

Fig. 6. Flow characteristic curves of stationary and sliding beds.

flow with a low-friction factor and stationary bed flow. Thefrictional pressure loss is presented inFig. 6 as a functionof the flow rate for both stationary and moving bed cases.The pressure loss for the stationary bed flow is higher thanthe sliding bed flow, because the stationary bed flow has rel-atively lower hydraulic diameter and higher flow velocity.An important feature of the sliding bed pressure loss curveis the presence of a minimum flow rate below which thebed becomes stationary. Hence, the sliding bed does not ex-ist below this flow rate. In order to get the overall picture,let us begin the cleaning at point A as shown inFig. 6 andincrease the flow rate from this point on. If we increase theflow rate so rapidly that the transport rate of the bed, whichincreases the hydraulic diameter, is negligible, then we willreach point B where the bed starts to slide over the chan-nel wall. Consequently, the flow mode becomes the slid-ing bed flow and the pressure loss has to follow the lowercurve. As a result, the pressure loss reduces to the value thatcorresponds to point C. Previous experimental and simula-tion studies (Doron and Barnea, 1993; Nguyen and Rahman,1998) confirmed a similar pressure loss reduction at the be-ginning of sliding. A further increase in the flow rate doesnot change the flow pattern. The pressure loss continuouslyincreases and follows the curve CD for higher flow rates.However, if the flow rate decreases after reaching point C,the curve follows the path CE, because the dynamic frictionfactor is lower than the static friction factor. Further decreasein flow rate stops the sliding of the bed and increases thepressure loss to the corresponding value of point F. The up-per curve begins to govern the course of the pressure loss.As indicated in the figure, the pressure curve may show ahysteresis depending on the static and dynamic friction fac-tors between the bed and the channel wall. When the flowrate reachesqmax, the stationary bed is at the onset of slid-ing. This condition can be extremely unstable. The stabilityof a bed mainly depends on the friction factor of the chan-nel, and roughness and thickness of the bed. When a con-stant pressure is maintained at the beginning of sliding, then

A. Ramadan et al. / Chemical Engineering Science 60 (2005) 2557–2570 2565

Table 1Common computational data for the model predictions

Particle density 2600 kg/m3 Channel diameter 70 mmDensity of water 1000 kg/m3 Static friction factor 0.5Density of PAC solution 1000 kg/m3 Dynamic friction factor 0.25Viscosity of water 0.001 Pa s

the flow rate abruptly increases fromqmax to qs . Therefore,the critical sliding condition can be one of the causes of theflow instability we have in stratified solid–liquid flows.

The model simulations were performed assuming thesame condition as the test. The average particle size wasused as the particle size. Common computational data forthe model predictions are listed inTable 1.

4. Experimental equipment and procedures

The experiments were conducted in a flow loop that hasa transparent test section with fluid re-circulation facilities.The flow loop is designed so that the flow is fully developedand free of any entrance effect at the test section. A simpli-fied schematic diagram of the loop is presented inFig. 7.As shown from the figure, the loop consisted of the follow-ing items: (i) test section (a 4-m long transparent pipe withinternal diameter of 70 mm); (ii) overhead tank to maintainconstant pressure head at the inlet of the channel; (iii) cir-culation tank for preparing and re-circulating the test fluids(water and PAC solutions); (iv) hydrocyclone to separate thesolids downstream of the channel; and (v) centrifugal pumpto recycle the fluid. The loop is equipped with a controlmechanism to maintain a more or less constant flow rate offluid during a test run. Details of the flow loop and equip-ment that were used in the experiment are available in pre-viously published studies (Ramadan, 2001; Ramadan et al.,2003).

The fluid level in the overhead tank was kept relativelyconstant in order to maintain steady-state flow condition inthe test section. Minimum and maximum level switches wereused to regulate the pump in accordance with the fluid levelin the overhead tank. A magnetic flow meter was placed

Test Section FlowMeter

Differencial PressureTransmitter

Hydrocyclone

Sand

OverheadTank

Control Valve

Centrifugal Pump

Mixing & Recirculation Tank

DP

FR

Fig. 7. Schematic diagram of the flow loop.

upstream of the channel to measure the flow rate. The pres-sure loss across the test section was continuously measuredusing a differential pressure transmitter. Both the flow me-ter and differential pressure transmitter were connected to apersonal computer for on-line display and recording. Tem-perature measurements of the fluid were made at the circu-lation tank. A manually operated valve, placed downstreamof the hydrocyclone, was used to regulate the flow rate.

The average transport rate tests were performed by placingthe sand sample uniformly in the test section and measuringthe time of complete removal of the sample, while keepingthe flow rate constant. Critical flow rate (critical velocity)tests were performed before every transport rate test. Thecritical flow rate was measured by visual observation ofthe movement of the bed particles (Ramadan et al., 2003).Rheologies of the PAC solutions were measured before everytest and maintained atK=0.050 Pa s0.7 andn=0.7. Duringthe test run, the temperature of the test fluid was maintainedat 20◦C by placing a heating element inside the circulationtank.

5. Result and discussion

The conditions for the experiments were selected to coverthe stationary bed condition at different flow rates (flow ve-locities). Physical characteristics of the sand beds and mea-sured critical flow rates are presented inTable 2(Ramadanet al., 2003). Figs. 8a–d present measured average transportrate and model predictions for water. Both the measured andmodel predicted transport rates show a generally increasingtrend with the flow rate. The model predictions show a sat-isfactory agreement with the measured data for the coarsesand beds (S3 and S4) regardless of the bed thickness andangle of inclination. As the flow rate approaches the criti-cal flow rate, the model prediction becomes inadequate forthe fine sand beds (S1 and S2). During the fine sand bedtests, we have also observed the formation of dunes and rip-ples.Henderson (1966)suggested that the transport mech-anisms of solids depend greatly on the difference betweenthe critical velocity and the mean flow velocity, and the par-ticle size. As the difference between these two flow ratesdecreases, there is a great possibility of dune and ripple for-mation. This phenomenon predominantly occurs when thebed particles are less than the viscous sublayer. The forma-tion of dunes and ripples considerably affects the intensityand modes of solids and cutting transport. As a result, thethree-layer model, which assumes a uniform bed thickness,fails to predict the transport rate near the critical flow rate.The grain Reynolds number is often used to describe thepossibilities of dune and ripple formation in sediment trans-port (FredsZe and Deigaard, 1992). After simplification, ageneralized form of the grain Reynolds number is written as

ReG = �LU2�

�o, (30)

2566 A. Ramadan et al. / Chemical Engineering Science 60 (2005) 2557–2570

Table 2Average critical flow rates of water and PAC solution with 1-litter sand bed in a horizontal channel

Particle Particle size Mean particle Critical velocity (m/s) Range ofRe (104) Range ofReG Critical flow rate(10−3 m3/s)size code range (mm) size (mm)

PAC Water PAC Water PAC Water PAC Water

S1 0.125–0.50 0.38 0.51 0.26 0.43–0.70 1.80–3.47 3–4 9–17 1.83 0.92S2 0.50–1.20 0.85 0.68 0.32 0.43–0.66 1.95–3.32 4–7 35–42 2.43 1.14S3 2.00–3.50 2.75 0.54 0.50 0.33–0.66 2.94–3.40 9–17 130–230 1.95 1.81S4 4.50–5.50 5.00 0.54 0.56 0.27–0.43 3.23–5.60 13–25 230–405 1.93 2.02

Flow Rate [10-3 m3/s]

1.5 2.0 2.5 3.0

Tra

nspo

rt R

ate

[Kg/

min

]

0.1

1

S1 ModelS2 ModelS3 ModelS1, ReG=9-14

S2, ReG=22-35

S3, ReG=131-155

Flow Rate [10-3 m3/s]

1.5 2.0 2.5 3.0

Tra

nspo

rt R

ate

[Kg/

min

]

0.1

1

S1 ModelS2 ModelS3 ModelS1, ReG=10-13

S2, ReG=23-44

S3, ReG=150-178

(a)

(c) (d)

(b)Flow Rate [10-3 m3/s]

1.5 2.0 2.5 3.0 3.5 4.0

Tra

nspo

rt R

ate

[Kg/

min

]

0.01

0.1

1

S1 ModelS2 ModelS3 ModelS4 ModelS1, ReG=9-17

S2, ReG=35-42

S3, ReG=130-230

S4, ReG=230-405

Flow Rate [10-3 m3/s]

1.5 2.0 2.5 3.0T

rans

port

Rat

e [K

g/m

in]

0.1

1

S1 ModelS2 ModelS3 ModelS1, ReG=10-12

S2, ReG=24-37

S3, ReG=140-166

Fig. 8. Measured and predicted average transport rates as a function of total flow rate for water: (a) 1-litter sand bed and� = 90◦; (b) 2.5-litter sand bedand� = 90◦; (c) 4-litter sand bed and� = 90◦; (d) 1-litter sand bed and� = 78◦.

where �o is the shear stress, which is determined by therheological model of the fluid at the representative shearrate,U�/dp. The ranges of grain Reynolds numbers for thesand beds are presented inTable 2. The grain Reynoldsnumber increases with increasing bed particle size. Modelpredictions are reasonable at high grain Reynolds numbers.Figs. 9a–d present the frictional pressure losses measuredat the beginning of the transport rate test and the modelpredictions for water test. During the test, the flow was fullyturbulent with the Reynolds number ranging from 18,000 to56,000. In general, the frictional pressure loss predictionsshow a satisfactory agreement with the measured data.

In order to determine the effect of rheology, transport ratemeasurements and simulations were performed using PACsolutions. The tests were performed at low Reynolds num-bers that were ranging from 2700 to 7000. Model predictedaverage transport rates are presented inFig. 10a togetherwith the measured ones. For the fine sand beds (S1 and S2),the figures indicated that the model predictions are consider-

ably higher than the measured data. For PAC solutions, thethickness of viscous sublayer is greater than that of waterbecause of its high viscosity. Hence, the viscous sublayercompletely buries protruding particles of the fine beds (S1and S2) forming dunes and ripples. Dunes and ripples wereformed during the experiment with the fine sand beds. As aresult, the model prediction becomes unsatisfactory for thefine sand beds at low flow rates. The discrepancies betweenthe predicted and measured transport rates increase as theflow rate and grain Reynolds number decrease. For coarsesand beds (S3 and S4), the model predictions are accept-able, although the accuracy the model reduces as the grainReynolds number decreases.

Fig. 10b presents pressure loss measurements along withthe model predictions for the PAC solution. For the finesand beds (S1 and S2), the model predictions are consid-erably higher than the measured values. Model predictionsfor coarse sand beds (S3 and S4) are relatively acceptable,although they have some discrepancies with the measured

A. Ramadan et al. / Chemical Engineering Science 60 (2005) 2557–2570 2567

Flow Rate [10-3 m3/s]

1.5 2.0 2.5 3.0Fric

tiona

l Pre

ssur

e Lo

ss [P

a/m

]

10

100

S1 ModelS2 ModelS3 ModelS1 MeasuredS2 MeasuredS3 MeasuredRe=2.8 - 5.0 E+04

Flow Rate [10-3 m3/s]

1.5 2.0 2.5 3.0Fric

tiona

l Pre

ssur

e Lo

ss [P

a/m

]10

100

S1 ModelS2 ModelS3 ModelS1 MeasuredS2 MeasuredS3 MeasuredRe=3.0 - 4.5 E+04

1.5 2.0 2.5 3.0Fric

tiona

l Pre

ssur

e Lo

ss [P

a/m

]

10

100

S1 ModelS2 ModelS3 ModelS1 MeasuredS2 MeasuredS3 MeasuredRe=2.8 - 4.8 E+04

1.5 2.0 2.5 3.0 3.5 4.0Fric

tiona

l Pre

ssur

e Lo

ss [P

a/m

]

10

100S1 ModelS2 ModelS3 ModelS4 ModelS1 MeasuredS2 MeasuredS3 MeasuredS4 MeasuredRe=1.8 - 5.6 E+04

(d)(c)

Flow Rate [10-3 m3/s] Flow Rate [10-3 m3/s](b)(a)

Fig. 9. Measured and predicted pressure gradient as a function of total flow rate for water: (a) 1-litter sand bed and� = 90◦; (b) 2.5-litter sand bed and� = 90◦; (c) 4-litter sand bed and� = 90◦; (d) 1-litter sand bed and� = 78◦.

Flow Rate [10-3 m3/s]

2.0 2.5 3.0 3.5 4.0

Tra

nspo

rt R

ate

[Kg/

min

]

0.01

0.1

1

S1 ModelS2 ModelS3 ModelS4 ModelS1, ReG=3-4S2, ReG=4-7 S3, ReG=9-17S4, ReG=13-25

(a)

(b) Flow Rate [10-3 m3/s]

2.0 2.5 3.0 3.5 4.0Fric

tiona

l Pre

ssur

e Lo

ss [P

a/m

]

100

1000

S1 ModelS2 ModelS3 ModelS4 ModelS1 MeasuredS2 MeasuredS3 MeasuredS4 MeasuredRe=2.7 - 7.0 E+03

Fig. 10. Measured and predicted average transport rates and pressuregradient as a function of total flow rate for PAC with 1-litter sand bedand � = 90◦: (a) average transport rates as a function of total flow rate;and (b) pressure gradient as a function of total flow rate.

data. Model predictions for the PAC solution are not as goodas that of water. The reasons for this can be the rheologyof the fluids, Reynolds number and flow characteristics ofthe polymer. According to Eq. (4), frictional pressure loss isa function of the wall shear stress and the interfacial shearstress. These two shear stresses mainly depend on the flowvelocity and friction factor. For water test, the flow was mostlikely in the turbulent regime with Reynolds number greaterthan 10,000. The PAC solution tests were conducted in thetransition regime. The accuracy of the friction factor cor-relations, Eq. (6), is relatively low in the transition regime,even though it is applicable for both turbulent and transitionregimes. In addition, the PAC solution to some extent ex-hibits drag-reducing behavior, which may affect the velocityprofile, friction factor and the flow regime.

To show the effect of angle inclination on the transportrate, theoretical simulations were made for 1-litter sand bedswith different particle sizes (S1 and S4) at different inclina-tion angles. Results of the simulations are presented inFigs.11a and b for water and PAC, respectively. The transportrate decreases as the inclination angle increases. The figuresshow the presence of two distinct parts in the transport ver-sus inclination graph. At lower inclination angles, there is ahigh-gradient part that suggests strong influence of the angleof inclination. The second part is a low-gradient part, whichis shown at higher inclination angles. In this part, the effectof inclination on the transport rate can be neglected. There-fore, the no-slip assumption, used byNguyen and Rahman(1998), which neglects the effect of inclination angle, is ac-ceptable in the low-gradient part.

2568 A. Ramadan et al. / Chemical Engineering Science 60 (2005) 2557–2570

Angle of Inclination [degrees]

30 40 50 60 70 80 90

Tra

nspo

rt R

ate

[Kg/

min

]

1

2

3

4

5Q=0.0017 m3/s S4Q=0.0025 m3/s S4Q=0.0033 m3/s S4Q=0.0017 m3/s S1Q=0.0025 m3/s S1Q=0.0033 m3/s S1

Angle of Inclination [degrees]

30 40 50 60 70 80 90

Tra

nspo

rt R

ate

[Kg/

min

]

1

2

3

4

5

(a)

(b)

Q=0.0017 m3/s S4Q=0.0025 m3/s S4Q=0.0033 m3/s S4Q=0.0017 m3/s S1Q=0.0025 m3/s S1Q=0.0033 m3/s S1

Fig. 11. Model predictions of the transport rate as a function of angle ofinclination for different sand beds: (a) water and (b) PAC.

The effect of particle size on the transport rate as the in-clination angle and flow rate change is also shown inFigs.11a and b. For both fluids, the influence of particle size onthe transport rate is strong at high flow rates and low incli-nation angles, and the coarse sand bed has a higher transportrate than the fine bed. Cutting transport with water appearsmore sensitive to the particle size. It is also worthwhile tocompare model-predicted cutting transport capacities of wa-ter and PAC. The model predictions indicated that, at a givenflow rate, PAC has better cutting transport capacity than wa-ter for this particular simulation condition. One possible ex-planation for this could be the rheology of the PAC solutionwith high consistency index (i.e., 0.052 Pa sn). Even thoughthe PAC solution has a shear thinning behavior, it is moreviscous than water even at higher shear rates. As a result,the hydrodynamic force (drag force) acting on particles ata give local fluid velocity becomes relatively high for thePAC solution. In addition to this, higher consistency indexreduces particle settling velocity of the suspended bed par-ticles. The reduction in the settling velocity has a positiveeffect in the cutting transport process. Therefore, because ofthese two phenomena, the PAC solution shows better cuttingtransport capacity than water. However, the PAC solutionrequires higher critical velocity (Table 2) than water for thefine beds, because the viscous sublayer tends to be thickerfor the PAC solution.

Flow Rate [10-3 m3/s]

0

2

4

6

8

10

12

S1S2S3S4

Flow Rate [10-3 m3/s]

0

2

4

6

8

10

12S1S2S3S4

2.0 2.4 2.8 3.2 3.6 4.0

2.0 2.4 2.8 3.2 3.6 4.0τ di

s/τdb

τ dis/τ

db

(a)

(b)

Fig. 12. Model predictions of the ratio of equivalent dispersive shear stress(�dis) to the fluid shear stress(�db) as a function of total flow rate in ahorizontal channel and for different sand beds: (a) water and (b) PAC.

Comparisons are made between the equivalent dispersiveshear stress (�dis) and the fluid shear stress (�db) using thesimulation results.Figs. 12a and b present the ratio of thesetwo shear stresses as a function of the flow rate for water andPAC, respectively. The figures reveal that the contribution ofthe dispersive shear stress in the momentum exchange be-tween the dispersed layer and the bed is considerably higherthan that of the fluid shear stress. The ratio decreases as theparticle size increases due to the increase in the friction fac-tor that directly influences the fluid shear stress. Especiallyfor water, the ratio is strongly influenced by the particle sizethat determines the bed roughness. The effect of bed rough-ness on the friction factor is not as significant for PAC solu-tion as for water. This is partly due to the fact that the PACsolution flow lies in the low Reynolds number range, andpartly due to the fact that the viscous sublayer tends to bethicker for the PAC solution. In turbulent flows, the natureof the flow is affected by the bed particle size (wall rough-ness). The influence of bed particle size can be effective aslong as the viscous sublayer near the bed is thin compared tothe particle size. This mostly happens at increased Reynoldsnumber. In practical applications, turbulent flows of non-Newtonian fluids usually fall in the low Reynolds numberrange. As a result, for non-Newtonian fluids such as poly-mer solutions (PAC solutions), the influence of the particlesize on the cutting transport rate is relatively less.

A. Ramadan et al. / Chemical Engineering Science 60 (2005) 2557–2570 2569

Fig. 12a indicates that the influence of the flow rate onthe stress ratio is insignificant for water. This means thatthe dispersive shear stress and the fluid shear stress increasein the same magnitudes as flow rate increases, resulting inapproximately constant stress ratio. However, for the PACsolution, the ratio increases significantly as the flow rateincreases. In this case, the relative change in fluid shear stresswith the increase in flow rate is less, because the flow isin the transition regime and the friction factor significantlydecreases as flow rate increases.

6. Conclusion

A three-layer solid transport model is developed and ap-plied to simulate solid particle transport in inclined chan-nels. The results from a series of experiments are com-pared with the model predictions. The data obtained clearlydemonstrated the limitations and application ranges of themodel. Consequently, the present formulation of the three-layer model can be applicable to predict the transport rateof stationary beds for both Newtonian and power-law fluidsif the flow rate is not close to the critical flow rate and thegrain Reynolds number is between 15 and 400. This meansthat when the eroding fluid velocity is relatively high overstationary beds of coarse particles, the model is applicablefor both Newtonian and power-law fluids. The model predic-tions for different angles of inclination indicate that the no-slip assumption in the formulation of the three-layer modelis acceptable at high inclination angles (horizontal and near-horizontal).

Notation

Ab cross-section area of the bedAd cross-section area of the dispersed layerAs cross-section area of the suspended layerAt flow cross-section area of the channelc concentrationcb mean volumetric concentration of solids in the

bedcd mean volumetric concentration of solids in dis-

persed layercs mean volumetric concentration of solids in the

suspended layercsl local volumetric concentration of solids in the

suspended layercave average concentration of solidsdp mean diameter of bed particlesdp/dx pressure gradientD pipe diameterDh hydraulic diameter of a layerfdb interfacial fluid friction factor between the bed

and dispersed layer

fs friction factor between the channel and sus-pended layer

Fb dry friction force between the bed and channelwall

Fd dry friction forces on the boundaries of the chan-nel wall and dispersed layer

Fdb dispersive forceFdyn dynamic friction force between the bed and

channel wallFst static friction force between the bed and channel

wallg gravitational accelerationh bed heighthd thickness of the dispersed layerhs thickness of the suspended layerK fluid consistency indexKs consistency index of the suspended layern flow behavior indexQ total flow rate of suspension through the channelQb flow rate of the bedQd flow rate of the dispersed layerQs flow rate of the suspended layerRe Generalized Reynolds numberRep particleReynolds numberReG grain Reynolds numberSb perimeter of the bed in contact with the channel

wallSd perimeter of contact surface between the channel

and dispersion layerSdb perimeter of contact surface between the bed and

dispersion layerSi perimeter in contact between suspended and dis-

persed layersSs channel perimeter in contact with the suspended

layerTr transport rate of solids or volume flow rate of

solidsTave average transport rate of solids or average vol-

ume flow rate of solidsuave mean fluid phase velocity in the mean flowub mean flow velocity of the bedud mean fluid phase velocity in the dispersed layerus mean fluid phase velocity in the suspended layerUave mean flow velocity, which is defined as the total

flow rate divided by the cross-sectional area ofthe channel.

Uc critical velocityUd mean flow velocity of the dispersed layerUd,r local velocity of the dispersed layer relative to

the bedUs mean flow velocity of the suspended layerU� friction velocityvd average settling velocity of solid particles in the

dispersed layervs average settling velocity of solid particles in the

suspended layer

2570 A. Ramadan et al. / Chemical Engineering Science 60 (2005) 2557–2570

vave average settling velocity of solid particles in thetotal flow

V volume of sandVs initial sand volumey vertical distance from the mean bed level

Greek letters

� angle of inclination from vertical diffusion coefficient of solid particles� angular displacement�b angular bed thickness of the bed�d angular bed thickness for dispersed layer� von Karman constant�d dry dynamic friction coefficient between the bed

and the channel wall�L fluid viscosity�s static friction coefficient between the bed and

channel wall�p density of solid particles�s density of suspended layer�d density of dispersed layer�L fluid density� shear stress acting on settling particles at repre-

sentative shear rate�b interstitial fluid shear stress between the bed and

channel wall�d fluid shear stress between the channel wall and

suspended layer�db fluid shear stress between the bed and dispersion

layer�dis equivalent dispersive shear stress between the

bed and dispersed layer�i interfacial shear stress between the suspended

layer and dispersed layer�s interstitial shear stress between the channel and

suspended layer�D equivalent dynamic friction angle

Acknowledgements

The authors express their appreciation to the staff in theworkshops and laboratories at the Department of PetroleumEngineering and Applied Geophysics, NTNU for their assis-tance in building the flow loop and Statoil for the financialsupport of this project.

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